There are 18 female board members
and 12 male board members.
Notice how there are a total of 18 plus 12,
or 30, board members.
First question is how many ways are there
to make a committee of 10 board members?
When forming the committees, the order does not matter,
and therefore we can answer this question
using a combination.
Because there are 30 board members
and the committee is made up of 10 board members,
the number of ways to make a committee of 10 board members
from 30 board members is 30 choose 10.
We can write 30 choose 10 using this notation,
this notation,
or this notation.
Now let's evaluate this combination on the TI-84,
which uses this notation here.
We enter 30,
Math, ride it out to Probability,
select option three for combination,
then enter 10,
and Enter,
which gives us 30,045,015.
There are 30,045,015 ways
to make a committee of 10 board members
from the 30 board members.
Next question is how many ways are there
to make a committee of 10 board members
if exactly four must be female?
We need to be careful here because
if exactly four must be female,
because the committee consists of 10 board members,
that means exactly six must be male,
which means to determine the number of ways
to make a committee of 10 board members
if exactly four must be female,
we must determine two things,
the number of ways to select exactly four females
and the number of ways to select exactly six males
to make a committee of 10.
So this will be the number of ways to select four females.
This will be the number of ways to select six males.
And because there are 18 females
and the order doesn't matter,
the number of ways of selecting four females
from 18 females is 18 choose four,
again, because order doesn't matter.
And then we have times
the number of ways of selecting six males
from the 12 male board members, which is 12 choose six.
And now let's determine this product on the TI-84,
which is 2,827,440,
which means there are 2,827,440 ways
to make a committee of 10 board members
if exactly four must be female,
which also means exactly six must be male.
In the last question,
we're asked to determine the probability
of selecting a committee of 10 board members
where exactly four of the members were female,
which again also means exactly six of the members were male.
So we'll say the probability
of exactly four females,
and we know this is equal to
the favorable number of outcomes
divided by the total number of outcomes.
And the favorable number of outcomes is the number
of ways to make a committee of 10 board members
if exactly four must be female,
which is 2,827,440,
divided by the total of possible number of outcomes,
which is the number of ways to make a committee
of 10 board members from all 30 members,
which is 30,045,015.
Let's go ahead and find this quotient.
Again, 2,827,440
divided by 30,045,015,
which, to four decimal places, is approximately 0.0941,
which is 9.41%.
I hope you found this helpful.